Transcript Statistical Learning Methods

Statistical Learning Methods
Chapter 20 (20.1, 20.2, 20.3, 20.4)
Copyright, 1996 © Dale Carnegie & Associates, Inc.
Statistical Learning
Data – instantiations of some or all of the random
variables describing the domain; they are evidence
Hypotheses – probabilistic theories of how the
domain works
The Surprise candy example: two flavors in very
large bags of 5 kinds, indistinguishable from outside
(i.e., P(X|hi))
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h1: 100% cherry – P(c|h1) = 1, P(l|h1) = 0
h2: 75% cherry + 25% lime
h3: 50% cherry + 50% lime
h4: 25% cherry + 75% lime
h5: 100% lime
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Problem formulation
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Given a new bag, random variable H denotes the bag type (h1 –
h5); Di is a random variable (cherry or lime); after seeing D1, D2, …,
DN, predict the flavor (value) of DN-1.
Bayesian learning
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Calculate the probability of each hypothesis, given the data and
make predictions on that basis
 P(hi|d) = αP(d|hi)P(hi), where d are observed values of D
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Make a prediction about an unknown quantity X
 P(X|d)=iP(X|d,hi)P(hi|d)=iP(X|hi)P(hi|d)=iP(X|hi)P(d|hi)P(hi)/P(d)
assuming that hi determines a probability distribution over X
Predictions use a likelihood-weighted average over hypotheses
 Hypothesis prior P(hi) and likelihood of the data under each hi,
P(d|hi);
One distribution for P(hi) is <0.1, 0.2, 0.4, 0.2, 0.1>.
 hi are intermediaries between raw data and predictions
 No need to pick one best-guess hypothesis
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P(d|hi) = ΠjP(dj|hi) assuming observations are
i.i.d. (independent, and identically distributed)
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Useful in calculating P(hi|d)
How each P(hi|d) changes when a sequence of
10 lime candies is observed (Fig. 20.1a,b)
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True hypothesis eventually dominates the Bayesian
prediction – the feature of Bayesian learning
Bayesian prediction is optimal; its hypothesis space is
usually very large or infinite
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Let’s see how these curves (Fig.20.1 a) are obtained
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P(h2|l1) = ; P(h2|l1,l2) = ; P(h2|l1,l2,l3) = ; …
How to obtain Fig.20.1 b – P(X|d) for X = lime
 P(X|l1) = iP(X|hi)P(hi|d)= iP(lime|hi)P(hi|l1)=;
P(X|l1,l2) = ; …
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Common approximations
MAP (maximum a posteriori): P(X|d) ~ P(X|hMAP)
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P(lime|l1,l2,l3) = 0.8 (from Fig 20.1b), but after seeing l1,l2,and
l3, P(h5|l1,l2,l3) is the max, P(lime|h5)=1.
hMAP is hi that maximizes P(hi|d)  P(d|hi)P(hi) (Fig 20.1a)
 Finding MAP hypotheses is much easier than Bayes Learning
 B and MAP learning use the prior to penalize complexity
 MAP learning chooses hi that compresses data most, or minimum
description length (MDL)
ML (maximum likelihood) can be obtained from MAP if
P(hi) is uniform
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hML is hi that maximizes P(d|hi) (Why?)
It is reasonable when (1) no preferable hypothesis a priori, (2)
data set is large
In short, P(X|d)iP(X|hi)P(d|hi)P(hi) – (Bayes learning)
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 P(X|hMAP)P(d|hMAP)P(hMAP) – (MAP)
 P(X|hML)P(d|hML) – (ML)
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Learning with Complete Data
Parameter learning - to find the numerical parameters
for a probability model whose structure is fixed
Data are complete when each data point contains
values for every variable in the model
Maximum-likelihood parameter learning: discrete model
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Two examples: one parameter and three parameters
 To be seen in next 3 slides (From the book authors’ slides)
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With complete data, the ML parameter learning problem for a
Bayesian network decomposes into separate learning problems,
one for each parameter
A significant problem with ML learning – 0 may be assigned to
some events that have not been observed
 Various tricks are used to avoid this problem. One is to
assign count = 1 instead of 0 to each event
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Add `–’ in L()
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After optimization
So, we are convinced by the previous
optimization procedure
What’s the learning from data?
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Estimate the probabilities from data
The details can be seen in the example of
NBC
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Naïve Bayes models
The most common Bayesian network model used in
machine learning
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Figure 20.3 Comparison between DT and NBC
It assumes that the attributes are conditionally
independent of each other, given class
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Fig 20.2(b) is a NBC with the parameters for ith instance as:
Θ=P(C=T), Θi1=P(Xi=T|C=T), Θi2=P(Xi=T|C=F)
Θ=P(Cherry), Θi1=P(Red|Cherry), Θi2=P(Red|¬Cherry)
A deterministic prediction can be obtained by choosing
the most likely class
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P(C|x1,x2,…,xn) = αP(C) Πi P(xi|C)
NBC has no difficulty with nosy data
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For n Boolean attributes, there are just 2n+1 parameters, no
search is required for finding hML
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Learning with Hidden Variables
Many real-world problems have hidden variables
which are not observable in the data available
for learning.
Question: If a variable (disease) is not observed,
why not construct a model without it?
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A: we’re lazy, so we don’t want to do without it. (??)
Answer: Hidden variables can dramatically
reduce the number of parameters required to
specify a Bayesian network. This results in the
reduction of needed amount of data for learning.
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Fig. 20.7 from 708 parameters to 78.
Hidden variables do complicate the learning problem.
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EM: Learning mixtures of Gaussians
The unsupervised clustering problem (Fig 20.8 with 3
components): P(x) = ki=1P(C=i)P(x|C=i)
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If we knew which component generated each xj, we can get ,
If we knew the parameters of each component, we know to
which ci should xj belong. However, we do not know either, …
EM – expectation and maximization
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Pretend we know the parameters of the model and then to infer
the probability that each xj belongs to each component; iterate
until convergence.
For the mixture of Gaussians, initialize the mixture model
parameters arbitrarily; and iterate the following
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E-step: Compute pij = P(C=i|xj) = αP(xj|C=i)P(C=i)
P(C=i) = pi = j pij
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M-step: Compute the new i = j pijxj/pi, i pijxjxjT/pi, wi = pi
(component weight)
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E-step computes the expected value pij of the
hidden indicator variables Zij, where Zij is 1 if xj
was generated by i-th component, 0 otherwise
M-step finds the new values of the parameters
that maximize the log likelihood of the data,
given the expected values of Zij
Fig 20.8(c) is learned from Fig. 20.8(a)
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Fig 20.9(a) plots the log likelihood of the data as EM
progresses
 EM increases the log likelihood of the data at each iteration.
 EM can reach a local maximum in likelihood.
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EM: Learning Bayesian networks with
hidden variables
Fig 20.10: two bags of mixed candies described
by 3 features: Flavor, Wrapper, and Hole
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The candy distribution is described by a naïve Bayes:
the features are independent, given the bag
The parameters are: Θ – the probability that a candy
from bag 1; ΘF1 and ΘF2 – the probabilities that the
flavor is cherry given that a candy from bag 1 and
bag2; ΘW1 and ΘW2 for red wrapper from bag1 and
bag 2; and ΘH1 and ΘH2 that the candy has a hole
from bag1 and bag2.
EM details are …
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Instance-based Learning
Parametric vs. nonparametric learning
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Learning focuses on fitting the parameters of a
restricted family of probability models to an
unrestricted data set
Parametric learning methods are often simple and
effective, but can oversimplify what’s really happening
Nonparametric learning allows the hypothesis
complexity to grow with the data
IBL is nonparametric as it constructs hypotheses
directly from the training data.
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Nearest-neighbor models
The key idea: Neighbors are similar
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Density estimation example: estimate x’s probability
density by the density of its neighbors
Connecting with table lookup, NBC, decision trees, …
How to define neighborhood N
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If too small, no any data points
If too big, density is the same everywhere
A solution is to define N to contain k points, where k
is large enough to ensure a meaningful estimate
 For a fixed k, the size of N varies (Fig 21.12)
 The effect of size of k (Figure 21.13)
 For most low-dimensional data, k is usually between 5-10
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K-NN for a given query x
Which data point is nearest to x?
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We need a distance metric, D(x1, x2)
Euclidean distance DE is a popular one
When each dimension measures something different, it is
inappropriate to use DE (Why?)
Important to standardize the scale for each dimension
 Mahalanobis distance is one solution
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Discrete features should be dealt with differently
 Hamming distance
Use k-NN to predict
High dimensionality poses another problem
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The nearest neighbors are usually a long way away!
 A d-dimensional hypercube of side b  1, its volume is bd, given N
data points, for each neighborhood to have k points, how big is b?
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Summary
Bayesian learning formulates learning as a form of
probabilistic inference, using the observations to update
a prior distribution over hypotheses.
Maximum a posteriori (MAP) selects a single most likely
hypothesis given the data.
Maximum likelihood simply selects the hypothesis that
maximizes the likelihood of the data (= MAP with a
uniform prior).
EM can find local maximum likelihood solutions for
hidden variables.
Instance-based models use the collection of data to
represent a distribution.
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Nearest-neighbor method
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